COMSOL 6.4 - Eigenfrequency Analysis

Eigenfrequency Analysis

The eigenfrequency analysis solves for the eigenfrequency of a model. The time-harmonic representation of the fields is more general and includes a complex parameter in the phase:

where the eigenvalue, (−λ) = −δ + jω, has an imaginary part representing the eigenfrequency and a real part responsible for the damping. It is often more common to use the quality factor or Q factor, which is derived from the eigenfrequency and damping:

Variables Affected by Eigenfrequency Analysis

The following table lists the variables that the eigenfrequency analysis affects:


Name	Expression	Can be Complex	Description
omega	imag(-lambda)	No	Angular frequency
damp	real(lambda)	No	Damping in time
Qfact	0.5*omega/abs(damp)	No	Quality factor
nu	omega/(2*pi)	No	Frequency

Nonlinear Eigenfrequency Problems

For some combinations of formulation, material parameters, and boundary conditions, the eigenfrequency problem can be nonlinear, which means that the eigenvalue enters the equations in a form other than the expected second-order polynomial form. The following table lists those combinations:


Solve for	Criterion	Boundary condition
E	Nonzero conductivity	Impedance boundary condition
E	Nonzero conductivity at adjacent domain	Scattering boundary condition
E	Analytical ports	Port boundary condition

These situations may require special treatment, especially since they can lead to “singular matrix” or “undefined value” messages if not treated correctly. Under normal circumstances, the automatically generated solver settings should handle the cases described in the table above. However, the following discussion provides some background to the problem of defining the eigenvalue linearization point. The complication is not only the nonlinearity itself, but also the way it enters the equations. For example, the impedance boundary condition with nonzero boundary conductivity has the term

where (−λ) = −δ + jω. When the solver starts to solve the eigenfrequency problem, it linearizes the entire formulation with respect to the eigenvalue around a certain linearization point. By default, this linearization point is set to the value specified in the field for the three cases listed in the table above. Normally, this should be a good value for the linearization point. For instance, for the impedance boundary condition, this avoids setting the eigenvalue λ to zero in the denominator in the above equation. For cases other than those listed in the table above, the default linearization point is zero.

If the default values for the linearization point is not suitable for your particular problem, you can manually provide a “good” linearization point for the eigenvalue solver. Do this in the node (not the Eigenfrequency node) under the node in the branch of the Model Builder. A solver configuration can be generated first. In the section, select the checkbox and enter a suitable value in the field. For example, if it is known that the eigenfrequency is close to 1 GHz, enter the eigenvalue 1[GHz] in the field.

In many cases it is enough to specify a good linearization point and then solve the problem once. If a more accurate eigenvalue is needed, an iterative scheme is necessary:

Specify that the eigenvalue solver only searches for one eigenvalue. Do this either for an existing solver configurations in the node or, before generating a solver sequence, in the node.

Solve the problem with a “good” linearization point. As the eigenvalue shifts, use the same value with the real part removed from the eigenvalue or, equivalently, use the real part of the eigenfrequency.

Extract the eigenvalue from the solution and update the linearization point and the shift.

Repeat until the eigenvalue does not change more than a desired tolerance.

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For a list of the studies available by physics interface, see The Wave Optics Module Physics Interface Guide

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Studies and Solvers in the COMSOL Multiphysics Reference Manual


	Band-Gap Analysis of a Photonic Crystal: Application Library path Wave_Optics_Module/Gratings_and_Metamaterials/bandgap_photonic_crystal